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\begin{document}

\title{Operations Research}
\author{Final Exam}
\date{May 21, 2016}
\maketitle

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\begin{enumerate}

\item Use the graphical method to minimize
\begin{gather*}
  z = x + y
\end{gather*}
subject to
\begin{gather*}
  x^2 + y^2 \geq 1 \,, \\
  x, y \geq 0 \,.
\end{gather*}
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\item Paula is using the Pyomo notebook displayed on the next page to
optimize the shipment of containers, described by the decision
variables $x_{ij}$, for her company.

\begin{enumerate}
\item Which problem is she solving?  Describe the problem in words.
You may complete the model scenario in any way consistent with the
given information.

\item Which problem is she solving?  State the problem in traditional
mathematical notation.

\item A container must be shipped as a unit.  Paula remembers the
``integrality property'' from her OR class and is surprised by the
solution she is obtaining.  Can you explain what is going on?  How can
she obtain an integer solution?

\item Give a \emph{brief} interpretation of the program output shown
in cells \verb+[4]+ and \verb+[5]+.

\end{enumerate}
\hfill(5+5+5+5)



\item Use dynamic programming to solve the following \emph{integer}
nonlinear programming problem.
\begin{gather*}
  \text{Minimize } z = 3 \, x_1 + 2 \ x_2^2 + x_3^3 \\
  \text{subject to } x_1 + x_2 + x_3 = 4 \,, \\
  x_1, x_2, x_3 \geq 0 \,.
\end{gather*}
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\item Customs authorities have busted a smuggling ring specialized in
exclusive watches.  Authorities are now selling off the seized
merchandise for EUR 250 a piece.  It is known that the true value of
each hand-made watch is EUR 2000, but also that there are many cheap
replicas in circulation with a market value of only EUR 40 a piece.
The conditions of sale do not provide any warranty whether a watch is
genuinely hand-made.  Prior to purchase, only a visual inspection is
allowed.  A lay person will not be able to tell a genuinely hand-made
watch from a replica.

\begin{enumerate}
\item The trade press has recently reported that 90\% of all watches
circulating on the black market are replicas.  Given this information,
would you choose to acquire watches at the quoted price?

\item An expert is offering her services to do a visual pre-screening
of the watches.  In the past, she has been correct 75\% of the time.
Compute the probability that she will classify one of the watches as
genuinely hand-made.

\item Given that the expert classifies a watch as hand-made, what is
the probability that the watch is truly hand-made?

\item What is the maximal fee per watch you should be willing to pay
for her expert evaluation?

\end{enumerate}
\hfill(5+5+5+5)




\item An airplane manufacturer is contracted to produce a small number
of a particular type of airplane during the coming years.  The
manufacturer will need to decide each year whether to set up a
production run with a fixed set-up cost of EUR\,1\,000\,000 per run.
During each production run, the manufacturer can make at most 6
airplanes.  If an airplane is not delivered during the year it is
produced, it will incur a holding cost of EUR\,100\,000 per year.  The
number of airplanes required are $r_1=1$, $r_2=6$, $r_3=2$, and
$r_4=3$ during each of the years.

Which production schedule(s) minimize the total cost for setup and
storage? 
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\item Consider an inventory under the assumptions of the economic
order quantity (EOQ) model with replacement at a constant rate,
namely:
\begin{itemize}
\item Constant demand rate of $d$ units per time,
\item continuous review with no planned shortages,
\item an order with batch size $Q$ is used to replenish the inventory
at a constant rate $r>d$ units per time where replenishments and
withdrawals can happen simultaneously,
\item each order incurs a setup cost $K$ per batch,
\item items in the inventory incur a holding cost $h$ per unit per time.
\end{itemize}
Derive the economic order quantity $Q^*$ as a function of $r$, $d$,
$h$, and $K$.


\end{enumerate}
\hfill(20)

\end{document}